Seismic uplift capacity of shallow horizontal strip anchor under … · 2017-02-06 · Seismic uplift capacity of shallow horizontal strip anchor under oblique load using pseudo-dynamic

The Japanese Geotechnical Society

Soils and Foundations

Soils and Foundations 2013;53(5):692–707

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Seismic uplift capacity of shallow horizontal strip anchor under oblique loadusing pseudo-dynamic approach

Sunil M. Rangari, Deepankar Choudhuryn, D.M. Dewaikar

Department of Civil Engineering, Indian Institute of Technology (IIT) Bombay, Powai, Mumbai 400076, India

Received 11 January 2012; received in revised form 17 May 2013; accepted 18 June 2013Available online 24 September 2013

Abstract

In this paper, an analytical method to compute the uplift capacity of an obliquely loaded horizontal strip anchor under both static and seismicconditions is described using the limit equilibrium method. The distribution of the soil reactions on a simple planar failure surface is obtainedthrough the use of Kötter's equation, and the pseudo-dynamic approach is used to obtain the net seismic vertical uplift capacity factor for the unitweight component of the soil (Fγd). The results for the static and seismic vertical uplift capacity factors are determined for various combinationsof input parameters, such as the load inclination, the soil friction angle, the embedment ratio, the soil amplification and both horizontal andvertical pseudo-dynamic seismic accelerations. It is observed that the orientation of the load significantly affects the seismic uplift capacity of thehorizontal strip anchor. Fγd is seen to decrease with an increase in both horizontal and vertical seismic accelerations and soil amplification,whereas it is seen to increase with an increase in the embedment ratio and the soil friction angle, as expected. The results in terms of the non-dimensional net seismic uplift capacity factor are presented in graphical and tabular forms. The present results are compared and found to be ingood agreement with similar results available in literature.& 2013 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved.

Keywords: Horizontal strip anchor; Kötter's equation; Oblique load; Limit equilibrium method; Pseudo-dynamic approach; Soil amplification; Seismic upliftcapacity; Closed-form analytical solution

1. Introduction

Ground anchors are commonly used as foundation systems forimportant structures requiring uplift resistance, such as transmis-sion towers, pipe lines buried under water, sheet pile walls, etc.It is well known that horizontal anchors are subjected to the vertical

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uplift load, and that when it reaches the ultimate load condition,failure surfaces develop around the anchor plate. These failuresurfaces reach the ground surface for shallow anchors. Theproblem becomes more complex and important under seismicconditions due to the devastating nature of earthquake forces onsuch ground anchors. The importance of the computation of thevertical uplift capacity of horizontal ground anchors is clear; it isan important topic for geotechnical engineers to address, underboth static and seismic conditions. Hence, the effect of earth-quakes on the uplift capacity of a strip anchor is studied in thepresent theory, as this knowledge is vital to the study of thebehavior of strip anchors under seismic conditions.For static conditions, numerical solutions for the uplift

capacity of horizontal anchors were obtained using the limitequilibrium method (Vesic, 1971; Meyerhof and Adams, 1968).The finite element method was used with a limit analysis to

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S.M. Rangari et al. / Soils and Foundations 53 (2013) 692–707 693

compute the uplift capacity of anchors (Kumar and Kouzer,2008; Merifield and Sloan, 2006; Tagaya et al., 1983). Experi-mental results and analytical solutions were obtained using thelimit equilibrium method and a limit analysis (Rowe and Davis,1982; Murray and Geddes, 1987; Tagaya et al., 1988; Dickinand Laman, 2007). Analytical models were presented by someresearchers (Chattopadhayay and Pise, 1986; Deshmukh et al.,2011), whereas Kumar (1999) and Subba Rao and Kumar(1994) used an upper bound limit analysis to predict the pulloutcapacity of ground anchors, and Honda et al. (2011) employed atwo-dimensional distinct element analysis to obtain the upliftcapacity of belled and multi-belled piles in sand. Series ofcentrifuge tests were presented by Hugo et al. (2010) todetermine the effect of gapping on the uplift capacity of ashallow skirted foundation. Miyata et al. (2011) obtained theaccuracy of a single square anchor plate for obtaining thecapacity in a multi-anchor wall system by applying two models.

Fig. 1. The typical model considered under seismic conditions.

Fig. 2. Free body diagram with va

Few researchers have obtained the vertical uplift capacity ofhorizontal plate anchors under seismic conditions (Kumar,2001; Choudhury and Subba Rao, 2004; Rangari et al., 2011a).Based on field and laboratory tests, a semi-empirical relation

was developed for the breakout resistance of a horizontal stripanchor under an oblique load by Meyerhof (1973). Das andSeeley (1975), on the other hand, obtained model test resultsfor a square anchor in dry sand subjected to inclined loads andpresented in graphical form.A pseudo-dynamic approach, developed by Steedman and

Zeng (1990) and upgraded by Choudhury and Nimbalkar(2005), was used by Ghosh (2009) to obtain the seismic upliftcapacity of a horizontal plate anchor by considering the effectof seismic amplification using an upper bound limit analysisand assuming a simple planar failure surface.In a pseudo-static analysis, the dynamic loading induced by an

earthquake is considered as time-independent, which ultimatelyassumes that the magnitude and the phase of the acceleration areuniform throughout the depth of the soil layer. Also, in a pseudo-static analysis, the effect of the amplification of vibrations cannotbe considered; this effect generally occurs towards the groundsurface and depends on various soil parameters, such as dampingand the shear modulus of the soil (Steedman and Zeng, 1990).Rectifying these shortcomings of the pseudo-static approach,Steedman and Zeng (1990) proposed a pseudo-dynamicapproach, which considers the effect of the finite shear wavevelocity, horizontal seismic acceleration and the amplification ofvibrations, to obtain the active earth pressure on retaining wallsfor a particular value of soil friction angle. Later, Choudhury andNimbalkar (2005) modified the approach further to consider theeffect of vertical acceleration also due to vertically propagatingprimary waves along with horizontal seismic acceleration; they

rious forces on failure wedges.

S.M. Rangari et al. / Soils and Foundations 53 (2013) 692–707694

considered the effects of time, the phase difference and the effectof amplification in both shear and primary waves propagatingthrough the soil.

Kötter (1903) was used by researchers (Balla, 1961; Matsuo,1967, 1968) to obtain the soil resistance on the failure surface.Recently, the limit equilibrium method has been effectively appliedalong with Kötter's equation to obtain the vertical uplift capacity ofhorizontal strip anchors (Deshmukh et al., 2011; Rangari et al.,2011a). However, the use of Kötter (1903) for obtaining theseismic uplift capacity of horizontal strip anchors under an obliqueload is still scarce in literature and remains mostly unnoticed.

Hence, in this paper, an attempt has been made to obtain theuplift capacity of a horizontal strip anchor under both static andseismic conditions by employing Kötter's equation (1903) and bytaking the forces on the soil column above the anchor intoconsideration. It is well known that the problem of the computa-tion of the seismic uplift capacity of horizontal strip anchors is the

Fig. 3. Variation in Fγd for various values of kh and kv for φ¼þ101 for (a) ε¼

application of the seismic passive resistance on retaining walls forthe negative wall friction condition (Choudhury and Subba Rao,2002, 2005a; Rangari et al., 2013), whereas the case of thepositive wall friction condition is used for bearing capacityanalyses (Subba Rao and Choudhury, 2005; Choudhury andSubba Rao, 2005b). A pseudo-dynamic approach is used with thelimit equilibrium method for a simple planer failure surface toobtain the seismic uplift capacity of an anchor with negative wallfriction angle δ=2ϕ/3 (Meyerhof and Adams, 1968; Choudhuryand Subba Rao, 2004; Rangari et al., 2011a, 2011b, 2013), whereϕ is the soil friction angle.

2. Method of analysis

The net seismic uplift capacity factor for a shallow horizontalstrip anchor, subjected to an inclined load, is obtained using thelimit equilibrium method and assuming a simple planar failure

1, (b) ε¼3, (c) ε¼5 and (a) ε¼7 with f¼1.0, H/λ¼0.3 and H/η¼0.16.


surface and a pseudo-dynamic approach for seismic forces. Thesoil is assumed as a rigid cohesionless material satisfying theMohr–Coulomb failure criterion. It is assumed that the inputvalues for the soil parameters, such as unit weight γ and soilfriction angle ϕ, are not affected under seismic conditions. Thesmooth anchor plate is considered by disregarding its mass in theanalysis. The marginal effect (o4%) of the roughness of theplate anchor on the calculated uplift capacity of a shallowhorizontal strip anchor for all soil friction angles and embedmentratios was observed by Merifield and Sloan (2006); it justifies theassumption of the smooth anchor plate used in the present study.It is also assumed that the soil lying below the surface of theanchor does not offer any resistance to the uplift capacity.

Fig. 1 shows the model considered for the horizontalplate anchor with an oblique load under seismic conditions.A horizontal plate anchor (CD) of width B is embedded at a

Fig. 4. Variation in Fγd for various values of kh and kv for φ¼�101 for (a) ε¼

depth H from the horizontal ground surface. W is the weight offailure wedge ACDB, R1 and R3 are the reactions on planarfailure surfaces BD and AC, respectively, and Qh and Qv arethe horizontal and vertical inertia forces acting in horizontaland vertical directions, respectively.Kötter’s equation (1903) for the passive state of earth pressure

in a cohesionless soil is employed to obtain the distribution ofsoil reactions on a curved surface and can be expressed as

dp

dsþ2p tan ϕ

dα

ds¼ γ sin ðαþϕÞ ð1Þ

where

1,

dp¼differential reaction pressure on failure surfaceds¼differential length of failure surfaceϕ¼soil friction angle

(b) ε¼3, (c) ε¼5 and (a) ε¼7 with f¼1.0, H/λ¼0.3 and H/η¼0.16.


dα¼differential angleγ¼unit weight of soilα¼ inclination of tangent on failure surface at point ofinterest with horizontal.

Integrating Eq. (1) and using the boundary conditions onfailure surfaces BD and AC, reactions R1 and R3 can beobtained as follows (Rangari et al., 2011a):

R1 ¼12γH2 sin ðα1þϕÞ

sin 2α1ð2Þ

R3 ¼12γH2 sin ðα3þϕÞ

sin 2α3ð3Þ


Failure wedge ACDB, shown in Fig. 1, under the pseudo-dynamic approach, considers the propagation of finite shearwave velocity Vs ¼

ffiffiffiffiffiffiffiffiffiG=ρ

pand primary wave velocity Vp

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðGð2�2νÞÞ=ðρð1�2νÞÞ

p, where G is the shear modulus,

and ρ and ν are the density and Poisson's ratio of the soil medium.The time of lateral shaking, T ¼ 2π=ω, is considered in thepresent analysis, where ω is the angular frequency of excitation.A free body diagram, with the failure model considered for the

present analysis, is shown in Fig. 2. For the seismic accelerationusing the pseudo-dynamic method, both horizontal and verticalsinusoidal vibrations with amplitude of acceleration in thehorizontal direction, ah (¼khg, where kh is the horizontalcoefficient of the seismic acceleration and g is the accelerationdue to gravity), and in the vertical direction, av (¼kvg, where kv isthe vertical coefficient of the seismic acceleration), are considered.The forces acting on failure block CDFE, failure wedge ACE andfailure wedge BDF are shown in Fig. 2(a), (b) and (c), respectively.



Let us consider the free body diagram of wedge BDF (Fig. 2(c)), subjected to both horizontal and vertical sinusoidal vibrationswith amplitudes of acceleration ah1 (¼khg) and av1 (¼kvg),respectively. Steedman and Zeng (1990) proposed the equation ofacceleration at depth z and time t acting in the horizontal directionby considering the effect of the finite shear wave velocity in termsof sinusoidal motion which was given by

ahðz; tÞ ¼ khg sin ω t�H�z

Vs

� �ð4aÞ

Later, Nimbalkar and Choudhury (2007) modified the basicequation for seismic acceleration to consider the effect of verticalacceleration, also due to vertically propagating primary wavesalong with horizontal seismic acceleration, and considered theeffects of time and amplification (f) in both shear and primarywaves propagating though the soil. In the present study, therefore,seismic horizontal and vertical accelerations at any depth z andtime t below the ground surface with soil amplification factor f


can be expressed as

ah1ðz; tÞ ¼ 1þ H�z

Hðf�1Þ

� �ah1 sin ω t�H�z

Vs

� �ð4bÞ

av1ðz; tÞ ¼ 1þ H�z

Hðf�1Þ

� �av1 sin ω t�H�z

Vp

� �ð5Þ

The mass of the small incremental part of thickness dz at depthz from the ground surface in failure wedge BDF is given by

m1ðzÞ ¼γ

g

ðH�zÞtan α1

dz ð6Þ

The total weight (W1) of failure wedge BDF is derived fromEq. (6) and yields

W1 ¼12

γH2

tan α1ð7Þ



The horizontal seismic inertia force acting on soil wedge BDFcan be written as

Qh1 ¼m1ðzÞah1ðz; tÞ ¼Z H

0

γ

g

H�z

tan α1

� �1þ H�z

Hðf�1Þ

� �khg sin ω t�H�z

Vs

� �dz

ð8aÞ

Integrating Eq. (8a), the total horizontal seismic inertia forcecan be expressed as

Qh1 ¼γkh

tan α14π22λπH cos ωζþλ2½ sin ωζ� sin ωt��

þ γkhðf�1Þtan α14π3

2π2λH cos ωζþ2πλ2 sin ωζ

þ λ3

Hð cos ωt� cos ωζÞ

ð8bÞ

where λ¼ TVs is the wavelength of the vertically propagatingshear wave and ζ¼ t�ðH=VsÞ.


Similarly the vertical seismic inertia force acting on soilwedge BDF can be expressed as

Qv1 ¼m1ðzÞav1ðz; tÞ ¼Z H

0

γ

g

H�z

tan α1

� �1þ H�z

Hðf�1Þ

� �kvg sinω t�H�z

Vp

� �dz

ð9aÞ

After integrating Eq. (9a), the total vertical seismic inertiaforce, one can obtain

Qv1 ¼γkv

tan α14π2½2πηH cos ωψþη2ð sin ωψ� sin ωtÞ�

þ γkvðf�1Þtan α14π3

2π2ηH cos ωψþ2πη2 sin ωψ

þ η3

Hð cos ωt� cos ωψÞ

ð9bÞ

where η¼ TVp is the wavelength of the vertically propagatingprimary wave through the soil wedge and ψ ¼ t�ðH=VpÞ.



Using the force equilibrium equation in horizontal andvertical directions, the seismic passive resistance (Ppγd1) canbe expressed as

Ppγd1 ¼ ½R1 sin ðα1þϕþφÞ�Qh1 cos φ�W1 sin φ

þQv1 sin φ�= cos ðδþφÞ ð10Þ

Ppγd1 ¼ ½�R1 cos ðα1þϕþφÞ�Qh1 sin φ

þW1 cos φ�Qv1 cos φ�= sin ðδþφÞ ð11ÞConsider the free body diagram of failure wedge ACE (Fig. 2(a)).The forces acting on wedge ACE are the W3 weight of failurewedge ACE, and Qh3 and Qv3 are the seismic inertia forcecomponents in horizontal and vertical directions, respectively.

The horizontal seismic inertia force acting on failure wedgeACE is given by the following integral:

Qh3 ¼Z H

0

γ

g

H�z

tan α3

� �1þ H�z

Hðf�1Þ

� �khg sinω t�H�z

Vs

� �dz ð12aÞ


After integrating Eq. (12a), the total horizontal seismicinertia force can be expressed as

Qh3 ¼γkh

tan α34π2½2λπH cos ωζþλ2ð sin ωζ� sin ωtÞ�

þ γkhðf�1Þtan α34π3

�2π2λH cos ωζþ2πλ2 sin ωζ

þ λ3

Hð cos ωt� cos ωζÞ� ð12bÞ

Similarly, the vertical seismic inertia force is given by theintegral

Qv3 ¼Z H

0

γ

g

H�z

tan α3

� �1þ H�z

Hðf�1Þ


Vp

� �dz ð13aÞ

The total vertical seismic inertia force can be obtained byintegrating Eq. (13a) as

Qv3 ¼γkv

tan α34π2½2πηH cos ωψþη2ð sin ωψ� sin ωtÞ�



þ γkvðf�1Þtan α34π3

2π2ηH cos ωψþ2πη2 sin ωψ

þ η3

Hð cos ωt� cos ωψÞ

ð13bÞ

The seismic passive resistance (PPγd3) acting on face CE canbe obtained from the force equilibrium equation in horizontaland vertical directions as

Ppγd3 ¼ ½R3 sin ðα3þϕ�φÞþW3 sin φ

þQh3 cos φ�Qv3 sin φ�= cos ðδ�φÞ ð14Þ

Ppγd3 ¼ ½�R3 cos ðα3þϕ�φÞþW3 cos φ�Qh3

� sin φ�Qv3 cos φ�= sin ðδ�φÞ ð15ÞConsider the elementary strip of thickness dz at depth z from theground surface in rectangular failure block CDFE (Fig. 2(b)).


The mass of the elementary strip is given by

m2ðzÞ ¼γ Bdz

gð16Þ

The total weight of failure block CDFE, W2, is obtainedfrom Eq. (16), namely,

W2 ¼ γ BH ð17Þ

The total horizontal seismic inertia force acting on soil blockCDFE can be obtained as

Qh2 ¼Z H

0

γBdz

g1þ H�z

Hðf�1Þ

� �khg sin ω t�H�z

Vs

� �

¼ γBkh4π2

2πλð cos ωζ� cos ωtÞþ2πλðf�1Þ � cos ωζ

þ λ2ðf�1Þð sin ωζ� sin ωtÞH

ð18Þ



The total vertical seismic inertia force acting on soil blockCDEF can be given by

Qv2 ¼Z H

0

γBdz

g1þ H�z

Hðf�1Þ


Vp

� �

¼ γBkv4π2

2πηð cosωψ� cosωtÞþ2πηðf�1Þ cosωψ

þ η2ðf�1Þð sinωψ� sinωtÞH

ð19Þ

The seismic passive resistance acting on the imaginary faces ofretaining walls (Choudhury and Subba Rao, 2004), DF and CE,are Ppγd1 and Ppγd3, respectively. As mentioned by Meyerhof andAdams (1968), the anchor uplift capacity is an application of theretaining wall problem under the negative wall friction angle caseof passive earth pressure under the static condition due to the

Fig. 10. Variation in Fγd for various values of kh and kv for φ¼�201 for (a) ε

movement of the anchor block. Similar consideration was givenby Choudhury and Subba Rao (2004) for the pseudo-staticseismic analysis and applied in the present pseudo-dynamicanalysis for imaginary wall movement.The gross seismic pullout load (Pud) of an anchor is given by

Pud ¼ Ppγd1 sin ðδþφÞþPpγd3 sin ðδ�φÞþW2 cos φ�Qv2 cos φ�Qh2 sin φ ð20Þ

The net ultimate seismic uplift capacity of anchor qudnet isgiven by

qudnet ¼Pud�½W2 cos φ�Qv2 cos φ�Qh2 sin φ�

Bð21Þ

On simplification, qudnet can be written in the following form:

qudnet ¼ 0:5γBFγd ð22Þ

¼1, (b) ε¼3, (c) ε¼5 and (a) ε¼7 with f¼1.4, H/λ¼0.3 and H/η¼0.16.

Fig. 11. Effect of soil amplification factor on Fγd for different values of kh withkv¼0.0kh for ϕ¼300, φ¼þ101, ε¼3.0, H/λ¼0.3 and H/η¼0.16.


The net seismic uplift capacity factor in the above expression(Fγd) can be given as

Fγd ¼ ε2Kpγd½ tan ðδþφÞþ tan ðδ�φÞ��2ε sin φ tan ðδ�φÞ

þ kv sin φ

2π2B

2πηð cos ωψ� cos ωtÞþ2πηðf�1Þ cos ωψ

þ η2ðf�1Þð sin ωψ� sin ωtÞH

tan ðδ�φÞ

� kh cos φ

2π2B

2πλð cos ωζ� cos ωtÞþ2πλðf�1Þ

� cos ωζ þ λ2ðf�1Þð sin ωζ� sin ωtÞH

tan ðδ�φÞ

ð23Þwhere Kpγd is the net seismic passive earth pressure coefficientfor the unit weight component and the embedment ratio of theanchor, ε¼H=B.

The net seismic uplift capacity factor (Fγd) can be obtainedfrom the above expression for different combinations of loadinclination, soil friction angle, seismic acceleration coefficientsin both vertical and horizontal directions, soil amplificationfactor and embedment ratio of a shallow horizontal plateanchor subjected to an oblique load.

In the present analysis, the value of Vp=Vs ¼ 1:87 isconsidered by using the relationship between the primaryand the shear wave velocities with Poison's ratio of the soil(Das, 1993). It can be observed from Eq. (23) that Fγd is afunction of the angle of the failure plane, t/T, H/λ and H/η. Theminimum value for Fγd is obtained by optimizing Fγd withrespect to the angle of the failure planes and t/T.

Fig. 12. Variation in Fγd for various positive values of φ for kh¼0.2, kv¼0.0khfor ϕ¼400 with f ¼1.0, H/λ¼0.3 and H/η¼0.16.

2.1. Determination of critical angle of failure plane

The critical angle of failure planes depends on the soilfriction angle and remains constant under static conditions,whereas under seismic conditions, it shifts with changes in theinput parameters, like kh, kh and soil amplification in additionto ϕ. Hence, a trial and error procedure is used in the presentanalysis to find the critical angle of the failure planes fordifferent combinations of seismic acceleration coefficients inboth horizontal and vertical directions with the amplificationfactor and the inclination of load for various values of the soilfriction angle. The trial values obtained for α1 and α3 mustsatisfy the condition whereby the two computed values forPpγd1, from Eqs. (10) and (11), and Ppγd3, from Eqs. (14) and(15), are the same. The critical values for α1 and α3 are obtaineduntil a convergence is reached to the third decimal for variousvalues of ϕ with different combinations of kh, kv, φ and f.

Fig. 13. Variation in Fγd for various negative values of φ for kh¼0.2,kv¼0.0kh for ϕ¼400, with f¼1.0, H/λ¼0.3 and H/η¼0.16.

3. Results and discussion

The equation proposed in the case of cohesionless soil toavoid the phenomenon of shear fluidization for variouscombinations of kh and kv (Richards et al., 1990) and modifiedby Ghosh (2009) for the amplification effect, f, for the values

of ϕ, is considered and given by

ϕZ tan �1 f kh1�f kv

ð24Þ

The results for Fγd for a horizontal strip anchor subjected to anoblique load are presented in a combination of graphs andtables. The different input parameters, such as ϕ, φ, kh, kv, ε, fand their variations, are considered in the present study.Figs. 3 and 4 show the variations in Fγd with kh for various

combinations of kv and different values for ϕ, ε with f =1.0,H/λ=0.3 and H/η=0.16 for φ=þ101 and φ=�101, respectively.Similarly, Figs. 5 and 6 show the variations in Fγd with kh for


various combinations of kv, and different values for ϕ, ε withf=1.0, H/λ=0.3 and H/η=0.16 for φ=þ201 and φ=�201,respectively. Again, Figs. 7 and 8 are drawn for φ=7101,respectively, whereas Figs. 9 and 10 are plotted for φ=7201,respectively, to show the variations in Fγd with kh for variouscombinations of kv, and different values for ϕ, ε with f=1.4,H/λ=0.3 and H/η=0.16. Fig. 11 shows the effect of theamplification factor on Fγd for different values of kh withkv¼0.0kh for ϕ¼301, φ¼101, ε¼3.0, H/λ¼0.3 and H/η¼0.16.

It can be observed that Fγd marginally decreases with anincrease in the inclination of the anchor load up to þ201,and thereafter, Fγd increases with an increase in φ (Fig. 12).A significant increase in Fγd values with an increase in thenegative inclination of the load is observed (Fig. 13). Thevalues for Fγd decrease with an increase in the amplificationfactor and the inclination of the load in either direction for allthe combinations of seismic acceleration coefficients in bothhorizontal and vertical directions. For example, referring to

Table 1Values of Fγd for variation in φ for various values of kv and ϕ with kh¼0.1, ε¼5

ϕ kv φ¼�401 φ¼�301 φ¼�201 φ¼�

301 0.0kh 62.61 45.99 36.48 30.760.5kh 55.98 40.89 32.24 27.041.0kh 50.98 36.99 29.01 24.21

401 0.0kh 109.02 75.43 57.95 59.790.5kh 96.49 66.47 50.82 52.521.0kh 86.13 59.06 44.93 46.98

Table 2Values for inclination of failure planes for various values of kh with kv¼0.0kh, ε

kh 0.0 0.1 0.2α1( 1) 52.9 51.18 49.48

α3( 1) 52.9 54.61 56.3

Table 3Values for inclination of failure planes for various values of ϕ and f for kh¼0.1, ε

ϕ kv/kh f¼1.0

α1( 1)

101 0.0 56.030.5 67.311.0 74.82

201 0.0 53.550.5 59.911.0 65.62

301 0.0 51.180.5 55.601.0 59.87

401 0.0 48.900.5 52.251.0 55.65

Fig. 3, Fig. 4 and Table 1, for the case of ϕ¼300, kh¼0.1,kv¼0.0kh, ε¼5 and f¼1.4, it is observed that the net seismicuplift capacity factor decreases by 5.86% for φ¼þ101 andincreases by 10.76% for φ¼�101 if compared with the valuesfor Fγd for φ¼01 and the difference increases with an increasein kv and φ in both directions of load inclination. Again, fromFigs. 5 and 9, for the case of ϕ=401, ε=5, kh=0.2 andkv=0.5kh, it is seen that the value for Fγd with f=1.0 is 1.09times more than the value with f=1.4 for φ=þ201.It is observed that the inclinations of failure planes AC and

BD keep changing with changes in the soil friction angle, bothseismic acceleration coefficients in horizontal and verticaldirections and the amplification factor. From Table 2, it isseen that α3 increases and α1 decreases with an increase in khin the critical direction, as shown in Fig. 1. Similarly, both α3and α1 increase significantly with an increase in the seismicacceleration coefficient in the vertical direction. Referring toTable 3, for φ¼301 and ϕ¼401, kh¼0.1 and f¼1.0, α1 and α3

, f¼1.4, H/λ¼0.3 and H/η¼0.16.

101 φ¼01 φ¼101 φ¼201 φ¼301 φ¼401

27.45 25.93 25.87 27.19 29.9324.05 22.68 22.67 23.91 26.4721.46 20.22 20.23 21.42 23.84

41.77 38.48 37.26 37.79 40.0236.36 33.44 32.39 32.94 35.0231.88 29.28 28.37 28.93 30.88

¼1, ϕ ¼301, φ ¼101, f ¼1.0, H/λ¼0.3 and H/η¼0.16.

0.3 0.4 0.547.81 46.16 44.5758.01 59.65 61.27

¼3, φ¼301, H/λ¼0.3 and H/η¼0.16.

f¼1.4

α3( 1) α1

( 1) α3( 1)

59.01 55.30 59.6969.60 70.55 73.2975.89 78.4 79.48

56.80 52.85 57.5162.92 61.9 65.9068.05 69.34 72.14

54.61 50.46 55.3559.01 56.85 61.4962.98 62.88 66.88

52.51 48.15 53.2855.90 53 58.1559.18 57.91 62.80

Table 5Comparison of gross pullout capacity, Pud(N/m), with experimental results byMeyerhof (1973) under static condition (kh¼kv¼0.0).

φ Present study Meyerhof (1973)

01 202 228101 187 2160201 179 208301 177 205401 181 204


are 48.90 and 52.51 for kh¼0.0kh increases to 52.25 and55.90, respectively, for kh¼0.5kh. The inclination of the failureplanes further increases in the presence of the soil amplifica-tion factor resulting in the formation of steeper failure planes.It is worth noting that the embedment ratio does not affect thegeometry of the failure planes even under seismic conditions,which shows a resemblance to the observation under staticconditions, whereas the reverse observation was proposed byGhosh (2009). From Table 2, for ϕ¼301 and φ¼101, kh¼0.1,kv¼0.0, f¼1.0 and from Table 3 for ϕ¼301, φ¼301, kh¼0.1,kv¼0.0, f¼1.0 values of α1 and α3 are the same, i.e., 51.18and 54.61, respectively. It is also observed from Tables 2 and 3that the anchor inclination does not change the geometry of thefailure planes, which shows a resemblance to the prediction byDas and Seeley (1975).

Fig. 11 shows the effect of the amplification factor on thevalues for the net uplift capacity factor for ϕ¼301 withφ¼þ101, ε¼3.0, H/λ¼0.3 and H/η¼0.16 for different valuesof kh with kv¼0.0kh. It shows that Fγd decreases continuouslywith an increase in the soil amplification factor and the seismicacceleration coefficient in the horizontal direction. Referring toFig. 11, Fγd decreases by 1.22% if the amplification factor (f)changes from 1 to 1.2, 1.35% for a change in f from 1.2 to 1.4,1.47% for a change in f from 1.4 to 1.6, 1.62% when f changesfrom 1.6 to 1.8, 1.65% for a change in f from 1.8 to 2.0. FromFigs. 4–10, it is also observed that Fγd decreases with an increasein kv and f for all possible inclinations of load. Hence, properattention must be given by a geotechnical engineer to determinethe effect of soil amplification on Fγd as it may lead to thecatastrophic failure of anchors under seismic conditions. Pre-viously, very few researchers obtained the uplift capacity for anoblique loaded anchor under static conditions. The present studyshows the importance of obtaining the uplift capacity underseismic conditions considering the effect of amplification.

Fig. 14. Comparison of breakout factor (qud) with experimental results ofDickin and Laman (2007) with γ¼14.5 kN/m3, B¼1 m and ϕ¼351 understatic conditions (kh¼kv¼0.0).

4. Comparison of results

Table 4 shows a comparison of the breakout factor(qud ¼ Pud=AγH) in medium–dense sand under static condi-tions (kh¼kv¼0.0) for the test data of Das and Seeley (1975)with B¼0.064 m and γ¼14.71 kN/m3, ϕ¼311. The shapefactors proposed by Tagaya et al. (1988) are used for thecomparison. Up to an embedment ratio of 2, the proposedtheory follows the experimental results closely; thereafter, forε¼4.5 it over predicts the results (by 15.21%). The shape

Table 4Comparison of breakout factor (qud) with experimental results by Das and Seeley

φ Das and Seeley (1975)

ε¼1 ε¼2 ε¼4.5

01 1.53 2.08 3.29101 1.79 2.99 3.91201 2.44 3.92 4.37301 2.768 4.49 4.57401 3.23 4.84 4.86

factors used for the comparison are lower than the valuesrecommended by Meyerhof and Adams (1968) and higher thanthose recommended by Dickin and Laman (2007).The results of the present study and the test results obtained

by Meyerhof (1973) for ϕ¼431 for a gross pullout load of ahorizontal strip anchor subjected to an inclined load are givenin Table 5. The results obtained in the present study are infavorable comparison with the experimental results and followa similar trend, showing a constant decrease in the grosspullout load, except the marginal shift when φ changes from301 to 401. In the present study, the gross pullout loadincreases by 2.21% and decreases by 0.49% for Meyerhof(1973), when φ changes from 301 to 401.The experimental results of Dickin and Laman (2007), for

loose sand with B¼1 m, γ¼14.5 kN/m3, ϕ¼351 under staticconditions, are compared with the predictions of theoretical

(1975) under static condition (kh¼kv ¼0.0).

Present study

ε¼1 ε¼2 ε¼4.5

1.58 2.17 3.651.73 2.37 3.971.96 2.68 4.52.34 3.21 5.393.06 4.16 6.97


methods, as shown in Fig. 14. The solutions obtained byDeshmukh et al. (2011) and Kumar (2001), the lower boundresults by Merifield and Sloan (2006) and the proposed methodshow a favorable agreement with the experimental results up toε¼4.0; thereafter, they overpredict. The theory of Meyerhofand Adams (1968) overestimates the uplift capacity with theobserved one. For a horizontal strip anchor subjected tovertical load under static conditions, Merifield and Sloan(2006) quantify the effect of anchor roughness, which is lessthan 4%, and hence, justifies the assumption of the smoothplate in the present study.

Table 6 shows a comparison of qud with the available resultsin literature under seismic conditions for ϕ¼301, φ¼01,f¼1.0, ε¼1, H/λ¼0.3 and H/η¼0.16. It is observed that thepresent study shows a reasonable comparison with the resultsobtained by other researchers (Choudhury and Subba Rao,2004; Kumar, 2001; Ghosh, 2009) for ε=1 with variouscombinations for kh and kv. The difference in the resultsobtained in the present study and those by Choudhury andSubba Rao (2004) is found to vary about 1.28% to �5%,�4.16% to 1% with Kumar (2001).The ultimate seismic uplift capacity factor, FγE (¼Pud/γB

2),using the upper bound solution, was obtained by Ghosh (2009)for a horizontal strip anchor using the pseudo-dynamic approach.Table 7 shows a comparison of FγE, for which various values forkh and kv are considered for ϕ¼301, φ¼01, ε¼5, f¼1.0 and1.4, H/λ¼0.3 and H/η¼0.16. It is observed that the presentstudy predicts lower values than Ghosh (2009) for the givencombinations of kh and kv with various amplification factors. Thepercentage difference of the present theory, observed for f¼1.0ranges between �1.78% and �5.27% for kv¼0.0kh; �1.78%

Table 7Comparison of FγE for f¼1.0 and 1.4, ϕ¼301, φ¼01, ε¼5, H/λ¼0.3 and H/η¼0

kh Ghosh (2009)

f¼1.0 f¼1.4

kv¼0.0kh kv¼0.5kh kv¼0.0kh kv¼0.5kh

0.0 19.43 19.43 19.43 19.430.1 19.41 18.51 19.38 18.220.2 19.28 17.42 19.24 16.910.3 19.09 16.28 18.87 15.450.4 18.79 15.08 18.36 –

Note: ‘–’ shows the case of shear fluidization (Richards et al., 1990; Ghosh, 2009

Table 6Comparison of qud for ϕ¼301, φ¼01, f¼1.0, ε¼1, H/λ¼0.3 and H/η ¼0.16.

kh Present study Choudhury and S

kv¼0.5kh kv¼1.0kh kv¼0.5kh

0 1.56 1.56 1.540.1 1.44 1.37 1.450.2 1.32 1.18 1.350.3 1.20 1.00 1.26

to �6.15% for kv¼0.5kh for different values of kh and differenceincreases with an increase in f (¼1.4), with Ghosh (2009). Thisdifference in results can be attributed to the inherent differencesbetween the limit analysis method and the limit equilibriummethod, thus showing the adequacy of the present theory.

5. Conclusions

The net seismic uplift capacity factor for an obliquely loadedhorizontal strip anchor for the unit weight component of soil (Fγd)is computed using the limit equilibrium method with a pseudo-dynamic approach. Kötter (1903) is employed to obtain thedistribution of soil reaction on a simple planar failure surface. Itis observed that Fγd decreases for a certain positive inclination ofload (φffiþ201); thereafter, it increases and a significant increasein the results was observed for the inclination of load in a negativedirection under seismic conditions. Fγd decreases significantly withan increase in the seismic acceleration coefficients in both verticaland horizontal directions and the amplification factor and increaseswith an increase in the embedment ratio and the soil friction angle,as expected. The values obtained from the present analysis arecompared with the available specific results reported in literatureunder both static and seismic conditions and are found to be ingood agreement with them. Hence, Kötter (1903) worked as apowerful tool in the present analysis, namely, to obtain the upliftcapacity factor which shows the better agreement with the resultsavailable in literature, obtained by the pseudo-static, pseudo-dynamic approach, using various approaches like the complicatedupper bound limit method, the log-spiral failure surface and eventhe experimental results.

.16.

Present study

f¼1.0 f¼1.4

kv¼0.0kh kv¼0.5kh kv¼0.0kh kv¼0.5kh

19.09 19.09 19.09 19.0918.87 17.64 18.73 16.9318.60 16.41 18.24 15.1918.26 15.34 17.61 13.8117.85 14.46 16.82 –

).

ubba Rao (2004) Kumar (2001)

kv¼1.0kh kv¼0.5kh kv¼1.0kh

1.54 1.58 1.581.37 1.49 1.401.20 1.38 1.221.03 1.25 0.99


Nomenclature

ah amplitude of acceleration in the horizontal directionav amplitude of acceleration in the vertical directionB width of ground anchordp differential reaction pressure on failure surfacef amplification factorFγd net seismic uplift capacity factorFγE ultimate seismic uplift capacity factorH depth of anchor from groundG shear moduluskh seismic coefficient of acceleration in the horizontal

directionkv seismic coefficient of acceleration in the vertical

directionL length of anchorPud gross seismic pullout capacityPpγd1, Ppγd3 seismic passive resistance acting on an imaginary

faces of retaining wallqud breakout factorQh total horizontal inertia forceQv total vertical inertia forceR1,3 resultant soil reaction on failure planesS length of failure planet timeT period of lateral shakingVp primary wave velocityVs shear wave velocityW total weight of failure wedgeW1, W2 and W3 weights of respective failure wedgesz depth of point below ground levelϕ soil friction angleδ wall friction angleα inclination of tangent on failure surface at the point of

interest with horizontalα1, α3 angles of failure planes with horizontale embedment ratioγ unit weight of soilo angular frequencyρ density of soil mediumν Poisson's ratio of soil mediumλ wavelength of the vertically propagating shear waveη wavelength of the vertically propagating primary waveφ angle of pullout load with vertical.

References

Balla, A., 1961. The resistance to breaking out of mushroom foundations forpylons. In: Proceedings of the Fifth International Conference on SoilMechanics and Foundation Engineering, vol. I, Paris, France, pp. 569–576.

Chattopadhayay, B.C., Pise, P.J., 1986. Breakout resistance of horizontalanchors in sand. Soils and Foundations 26 (4), 16–22.

Choudhury, D., Nimbalkar, S., 2005. Seismic passive resistance by pseudodynamic method. Geotechnique 55 (9), 699–702.

Choudhury, D., Subba Rao, K.S., 2005a. Seismic uplift capacity of inclinedstrip anchors. Canadian Geotechnical Journal 42 (1), 263–271.

Choudhury, D., Subba Rao, K.S., 2005b. Seismic bearing capacity of shallowstrip footings. Geotechnical and Geological Engineering 23 (4), 403–418.

Choudhury, D., Subba Rao, K.S., 2004. Seismic uplift capacity of strip anchorsin soil. Geotechnical and Geological Engineering 22 (1), 59–72.

Choudhury, D., Subba Rao, K.S., 2002. Seismic passive resistance in soils fornegative wall friction. Canadian Geotechnical Journal 39 (4), 971–981.

Das, B.M., 1993. Principals of Soil Dynamics. PWS-KENT PublishingCompany, Boston, MA (Boston, Massachusetts).

Das, B.M., Seeley, G.R., 1975. Inclined load resistance of anchors in sand.Proceedings of the American Society of Civil Engineers 2 (GT 9),995–1003.

Deshmukh, V.B., Dewaikar, D.M., Choudhary, D., 2011. Uplift capacity ofhorizontal strip anchors in cohesionless soil. Geotechnical and GeologicalEngineering, vol. 29. Springer, Netherlands977–988.

Dickin, E.A., Laman, M., 2007. Uplift response of the strip anchor incohesionless soil. Advances in Engineering Software 38 (8–9), 618–625.

Ghosh, P., 2009. Seismic vertical uplift capacity of horizontal strip anchors usingpseudo-dynamic approach. Computers and Geotechnics 36 (1–2), 342–351.

Honda, T., Hirai, Y., Sato, E., 2011. Uplift capacity of belled and multi-belledpiles in dense sand. Soils and Foundations 51 (3), 483–496.

Hugo, E.A.-M., Gourvenec, S.M., Randolph, M.F., 2010. Effect of gapping onthe transient and sustained uplift capacity of a shallow skirted foundation inclay. Soils and Foundations 50 (5), 725–735.

Kötter, F., 1903. Die Bestimmung des Drucks an gekrummten Gleitflachen,eine Aufgabe aus der Lehre vom Erddruck. Sitzungsberichte der Akademieder Wissenschaften Berlin, 229–233.

Kumar, J., Kouzer, K.M., 2008. Vertical uplift capacity of horizontal anchorsusing upper bound limit analysis and finite elements. Canadian Geotechni-cal Journal 45 (5), 698–704.

Kumar, J., 2001. Seismic vertical uplift capacity of strip anchors. Geotechnique51 (3), 275–279.

Kumar, J., 1999. Kinematic slices approach for uplift analysis of strip anchors.International Journal for Numerical and Analytical Methods in Geomechanics23 (11), 1159–1170.

Matsuo, M., 1967. Study on uplift resistance of footing (I). Soils andFoundations 7 (4), 1–37.

Matsuo, M., 1968. Study on uplift resistance of footing (II). Soils andFoundations 8 (4), 18–48.

Merifield, R.S., Sloan, S.W., 2006. The ultimate pullout capacity of anchors infrictional soils. Canadian Geotechnical Journal 43 (8), 852–868.

Meyerhof, G.G., Adams, J.I., 1968. The ultimate uplift capacity of foundations.Canadian Geotechnical Journal 5 (4), 225–244.

Meyerhof, G.G., 1973. The uplift capacity of foundation under oblique load.Canadian Geotechnical Journal 10 (64), 64–70.

Miyata, Y., Bathurst, R.J., Konami, T., 2011. Evaluation of two anchor platecapacity models for MAW systems. Soils and Foundations 51 (5),885–895.

Murray, E.J., Geddes, J.D., 1987. Uplift of anchor plates in sand. Journal ofGeotechnical Engineering, ASCE 113 (3), 202–215.

Nimbalkar, S., Choudhury, D., 2007. Sliding stability and seismic design ofretaining wall by pseudo-dynamic method for passive case. Soil Dynamicsand Earthquake Engineering 27 (6), 497–505.

Rangari, S., Choudhury, D., Dewaikar, D.M., 2011a. Pseudo-static uplift capacityof horizontal strip anchor. In: Han, J., Alzamora, D.E. (Eds.), Geo-Frontiers2011: Advances in Geotechnical Engineering. Geotechnical Special Publica-tion No. 211, ASCE, Reston, VA, USA, pp. 1821–1831.

Rangari, S., Choudhury, D., Dewaikar, D.M., 2011b. Pseudo-static upliftcapacity of inclined strip anchor in cohesionless soil. Electronic Journal ofGeotechnical Engineering 16, 1185–1200.

Rangari, S., Choudhury, D., Dewaikar, D.M., 2013. Computations of seismicpassive resistance and uplift capacity of horizontal strip anchors in sand.Geotechnical and Geological Engineering 31 (2), 569–580.

Richards, R., Elms, D.G., Budhu, M., 1990. Dynamic fluidization of soils.Journal of Geotechnical Engineering, ASCE 116 (5), 740–759.

Rowe, R.K., Davis, E.H., 1982. The behaviour of anchor plates in sand.Geotechnique 32 (1), 25–41.

Steedman, R.S., Zeng, X., 1990. The influence of phase on the calculation ofpseudo-static earth pressure on a retaining wall. Geotechnique 40 (1), 103–112.

http://refhub.elsevier.com/S0038-0806(13)00089-9/othref0005



http://refhub.elsevier.com/S0038-0806(13)00089-9/sbref1











































































Subba Rao, K.S., Choudhury, D., 2005. Seismic passive earth pressures insoils. Journal of Geotechnical and Geoenvironmental Engineering 131 (1),131–135.

Subba Rao, K.S., Kumar, J., 1994. Vertical uplift capacity of horizontalanchors. Journal of Geotechnical Engineering 120 (7), 1134–1147.

Tagaya, K., Tanaka, A., Aboshi, H., 1983. Application of finite elementmethod to pullout resistance of buried anchor. Soils and Foundations 23(13), 91–104.

Tagaya, K., Scott, R.F., Aboshi, H., 1988. Pullout resistance of buried anchorin sand. Soils and Foundations 28 (13), 114–130.

Vesic, A.S., 1971. Breakout resistance of objects embedded in ocean bottom.Journal of Soil Mechanics and Foundation Engineering Division, AmericanSociety of Civil Engineers 97 (9), 1183–1205.














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Seismic uplift capacity of shallow horizontal strip anchor under … · 2017-02-06 · Seismic uplift capacity of shallow horizontal strip anchor under oblique load using pseudo-dynamic